Theorem iunxun 3709

Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iunxun	⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶)

Proof of Theorem iunxun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexun 3100	. . . 4 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶))
2		eliun 3635	. . . . 5 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶)
3		eliun 3635	. . . . 5 ⊢ (y ∈ ∪ x ∈ B 𝐶 ↔ ∃x ∈ B y ∈ 𝐶)
4	2, 3	orbi12i 668	. . . 4 ⊢ ((y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶) ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶))
5	1, 4	bitr4i 176	. . 3 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶))
6		eliun 3635	. . 3 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ ∃x ∈ (A ∪ B)y ∈ 𝐶)
7		elun 3061	. . 3 ⊢ (y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶) ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶))
8	5, 6, 7	3bitr4i 201	. 2 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶))
9	8	eqriv 2019	1 ⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶)

Syntax hints:   ∨ wo 616   = wceq 1228   ∈ wcel 1374  ∃wrex 2285   ∪ cun 2892  ∪ ciun 3631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-iun 3633

This theorem is referenced by:  iunsuc  4106  rdgisuc1  5891  oasuc  5959  omsuc  5966